Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

Nowadays Machine Learning (ML) techniques can provide new perspectives to identify hidden patterns and classes inside data. Applying ML to the Internet of Things (IoT) and its produced data represents a great challenge in every application domain, since analyzing IoT data increasingly requires the use of advanced mathematical algorithms, novel computational techniques and services. In this paper we present and discuss the application of unsupervised learning techniques on IoT data collected in a Cultural Heritage framework. Behavioural data have been gathered in a non-invasive way in order to achieve a ML classification that can be exploited by cultural stakeholders in terms of medium-long term strategy and also in terms of strictly operational decisions. The application of ML and other learning techniques will acquire a key role to complement the more traditional services with new intelligent ones able to satisfy the needs of companies, stakeholders and consumers.

Nowadays Unsupervised Learning can provide new perspectives to identify hidden patterns and classes inside the huge amount of data coming from the Internet of Things world. Analyzing IoT data through machine learning techniques requires the use of mathematical algorithms, computational techniques and an accurate tuning of the input parameters. In this paper we present a study of unsupervised learning techniques applied on IoT data to support decision making processes inside intelligent environments. To assess the proposed approach we discuss two case-of-study in which behavioural IoT data have been collected, also in a non-invasive way, in order to achieve an unsupervised classification that can be adopted during a decision making process. The use of Unsupervised Learning techniques is acquiring a key role to complement the more traditional services with new decision making ones supporting the needs of companies, stakeholders and consumers.

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.